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Theorem tgbtwntriv2 26848
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgbtwntriv2 (𝜑𝐵 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprl 768 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥))
2 tkgeom.p . . . . . 6 𝑃 = (Base‘𝐺)
3 tkgeom.d . . . . . 6 = (dist‘𝐺)
4 tkgeom.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (𝜑𝐵𝑃)
87ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵𝑃)
9 simplr 766 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝑥𝑃)
10 simpr 485 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → (𝐵 𝑥) = (𝐵 𝐵))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 26824 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵 = 𝑥)
1211adantrl 713 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 = 𝑥)
1312oveq2d 7291 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
141, 13eleqtrrd 2842 . 2 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15 tgbtwntriv2.1 . . 3 (𝜑𝐴𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 26825 . 2 (𝜑 → ∃𝑥𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵)))
1714, 16r19.29a 3218 1 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-trkgc 26809  df-trkgcb 26811  df-trkg 26814
This theorem is referenced by:  tgbtwncom  26849  tgbtwntriv1  26852  tgcolg  26915  legid  26948  hlid  26970  lnhl  26976  tglinerflx2  26995  mirreu3  27015  mirconn  27039  symquadlem  27050  outpasch  27116  hlpasch  27117
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